Excursion decomposition of the 2D continuum GFF

Juhan Aru, Titus Lupu, Avelio Sepúlveda

Published 2023-04-06Version 1

In this note we show that the 2D continuum Gaussian free field (GFF) admits an excursion decomposition similar to the classical excursion decomposition of the Brownian motion. In particular, 2D continuum GFF can be written as an infinite sum of disjoint positive and negative sign excursions, which are given by Minkowski content measures of clusters of a critical 2D Brownian loop soup with i.i.d. signs. Although the 2D continuum GFF is not even a signed measure, we show that the decomposition to positive and negative parts is unique under natural conditions.